Surface Area of Rectangular Prism: Express in Terms of X (3×5×X)

Q: Why do we multiply by 2 in the surface area formula?

A rectangular prism has 6 faces that come in 3 pairs of identical rectangles. Each pair has the same area (lw, lh, and wh), so we calculate the area of each unique face and multiply by 2.

Q: How do I remember which dimensions to multiply together?

Think of the three different ways to view the prism: length × width (top/bottom), length × height (front/back), and width × height (left/right sides). Each pair gives you one term in the formula.

Q: What if X equals a specific number like 4?

Simply substitute! If X = 4, then $ 16X + 30 = 16(4) + 30 = 64 + 30 = 94 $ square units. The algebraic expression lets you find surface area for any value of X.

Q: Can I use different letters instead of X?

Absolutely! Variables can be any letter. Whether it's X, t, n, or even d for depth, the process stays the same. Just substitute your variable into the surface area formula.

Q: How do I check if 16X + 30 is correct?

Work backwards: divide by 2 to get $ 8X + 15 $, then see if it equals $ (X)(5) + (X)(3) + (5)(3) = 5X + 3X + 15 = 8X + 15 $ ✓

Surface Area Formula with Variable Dimensions

Express the surface area of the rectangular prism in terms of X using the given data.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the surface area of the box

00:06 We'll use the formula for calculating rectangle area (side times side)

00:13 Calculate the area of each face

00:40 Since the box has 6 faces (2 of each type)

00:44 To calculate the surface area, multiply each area by 2 and sum

01:15 Group the terms

01:19 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Express the surface area of the rectangular prism in terms of X using the given data.

Step-by-step solution

To find the surface area of the rectangular prism in terms of $X$ , follow these steps:

Step 1: Identify the dimensions:
- Length $l = X$
- Width $w = 5$
- Height $h = 3$
Step 2: Use the surface area formula for a rectangular prism: $S = 2(lw + lh + wh)$
Step 3: Substitute the given values: $S = 2(X \cdot 5 + X \cdot 3 + 5 \cdot 3)$
Step 4: Simplify the expression: $S = 2(5X + 3X + 15)$ $S = 2(8X + 15)$ $S = 16X + 30$
Step 5: Therefore, the surface area of the rectangular prism expressed in terms of $X$ is $16X + 30$ .

This matches with choice 1.

Thus, the solution to the problem is $16X + 30$ .

Final Answer

16X+30

Key Points to Remember

Essential concepts to master this topic

Formula: Surface area = 2(lw + lh + wh) for rectangular prisms
Technique: Substitute X·5 + X·3 + 5·3 = 5X + 3X + 15
Check: Factor out 2(8X + 15) = 16X + 30 matches dimensions ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by 2 in the surface area formula
Don't calculate just lw + lh + wh = 8X + 15! This gives only half the surface area because each face appears twice on a rectangular prism. Always use the complete formula S = 2(lw + lh + wh) to account for all six faces.

Practice Quiz

Test your knowledge with interactive questions

Calculate the surface area of the orthohedron below using the data in the diagram.

FAQ

Everything you need to know about this question

Why do we multiply by 2 in the surface area formula?

A rectangular prism has 6 faces that come in 3 pairs of identical rectangles. Each pair has the same area (lw, lh, and wh), so we calculate the area of each unique face and multiply by 2.

How do I remember which dimensions to multiply together?

Think of the three different ways to view the prism: length × width (top/bottom), length × height (front/back), and width × height (left/right sides). Each pair gives you one term in the formula.

What if X equals a specific number like 4?

Simply substitute! If X = 4, then $16X + 30 = 16(4) + 30 = 64 + 30 = 94$ square units. The algebraic expression lets you find surface area for any value of X.

Can I use different letters instead of X?

Absolutely! Variables can be any letter. Whether it's X, t, n, or even d for depth, the process stays the same. Just substitute your variable into the surface area formula.

How do I check if 16X + 30 is correct?

Work backwards: divide by 2 to get $8X + 15$ , then see if it equals $(X)(5) + (X)(3) + (5)(3) = 5X + 3X + 15 = 8X + 15$ ✓

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